$sigma$-locales in Formal Topology

A σ-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that σ-frames, actually σ-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every σ-frame L is the lattice of Lindel ̈of elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over L. We then give a constructive characterization of the smallest (strongly) dense σ-sublocale of a given σ-locale, thus providing a “σ-version” of a Boolean locale. Our development depends on the axiom of countable choice.